Sampling Variability – The Heart of Inference
Salaries of football coaches
In your team…
First
- sample 10 salaries
- calculate the median
Then
- sample 25 salaries
- calculate the median
You have a sample of 25 UC & CSU coach salaries.
Would you feel comfortable inferring that the median salary of your sample is close to the median salary of all UC & CSU coaches?
Why or why not?
Why sample more than once?
Variability is a central focus of the discipline of Statistics!
Making decisions based on limited information is uncomfortable!
You likely weren’t willing to infer the population median salary from your sample!
Sampling Framework
population – collection of observations / individuals we are interested in
population parameter – numerical summary about the population that is unknown but you wish you knew
sample – a collection of observations from the population
sample statistic – a summary statistic computed from a sample that estimates the unknown population parameter.
Statistical Inference
There were 252 “Head Coaches” at University of California and California State Universities in 2019
| Median salary for all coaches |
|---|
| $137,619 |
Inferring information from your sample onto the population is called statistical inference.
Statistical Inference Reasoning
- If the sampling is done at random
- the sample is representative of the population
- any result based on the sample can generalize to the population
- the point estimate is a “good guess” of the unknown population parameter
Shouldn’t one random sample be enough then? Isn’t that what we use to make confidence intervals and do hypothesis tests?
Virtual Sampling
| Employee Name | Job Title | Total Pay & Benefits |
|---|---|---|
| Lyndsie Olson | Intercol Ath Head Coach Ex | 18075.0 |
| Nancy Mcdaniel | Head Coach 5 | 181157.0 |
| Sam Crosson | Head Coach 5 | 225184.0 |
| Lauren Beth Nadler | Head Coach 5 | 44714.0 |
| Sandra S Kriezel | HEAD COACH - 12 MONTH | 120699.5 |
| Charles Kelly | Intercol Ath Head Coach Ex | 3571805.0 |
\(\vdots\)
Distribution of 1000 medians from samples of 25 coaches
Sampling Distributions
- Visualize the effect of sampling variation on the distribution of any point estimate
- In this case, the sample median
- We can use sampling distributions to make statements about what values we can typically expect.
Be careful! A sampling distribution is different from a sample’s distribution!
Distributions of 1000 medians from different sample sizes
What differences do you see?
Variability for Different Sample Sizes
| Sample Size | Standard Error of Median |
|---|---|
| 25 | 19273.031 |
| 50 | 13018.543 |
| 100 | 7849.817 |
Standard errors quantify the variability of point estimates
As a general rule, as sample size increases, the standard error decreases.
Careful! There are important differences between standard errors and standard deviations.
A good guess?
Precision & Accuracy
- Random sampling ensures our point estimates are accurate.
- Larger sample sizes ensure our point estimates are precise.
Sampling Activity!